HomeHome Intuitionistic Logic Explorer
Theorem List (p. 35 of 102)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 3401-3500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsnidb 3401 A class is a set iff it is a member of its singleton. (Contributed by NM, 5-Apr-2004.)
 |-  ( A  e.  _V  <->  A  e.  { A } )
 
Theoremsnid 3402 A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 31-Dec-1993.)
 |-  A  e.  _V   =>    |-  A  e.  { A }
 
Theoremvsnid 3403 A setvar variable is a member of its singleton (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  x  e.  { x }
 
Theoremelsn2g 3404 There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. This variation requires only that  B, rather than  A, be a set. (Contributed by NM, 28-Oct-2003.)
 |-  ( B  e.  V  ->  ( A  e.  { B }  <->  A  =  B ) )
 
Theoremelsn2 3405 There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. This variation requires only that  B, rather than  A, be a set. (Contributed by NM, 12-Jun-1994.)
 |-  B  e.  _V   =>    |-  ( A  e.  { B }  <->  A  =  B )
 
Theoremmosn 3406* A singleton has at most one element. This works whether  A is a proper class or not, and in that sense can be seen as encompassing both snmg 3486 and snprc 3435. (Contributed by Jim Kingdon, 30-Aug-2018.)
 |- 
 E* x  x  e. 
 { A }
 
Theoremralsnsg 3407* Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)
 |-  ( A  e.  V  ->  ( A. x  e. 
 { A } ph  <->  [. A  /  x ]. ph )
 )
 
Theoremralsns 3408* Substitution expressed in terms of quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.)
 |-  ( A  e.  V  ->  ( A. x  e. 
 { A } ph  <->  [. A  /  x ]. ph )
 )
 
Theoremrexsns 3409* Restricted existential quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.) (Revised by NM, 22-Aug-2018.)
 |-  ( E. x  e. 
 { A } ph  <->  [. A  /  x ]. ph )
 
TheoremrexsnsOLD 3410* Restricted existential quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.) Obsolete as of 22-Aug-2018. Use rexsns 3409 instead. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( A  e.  V  ->  ( E. x  e. 
 { A } ph  <->  [. A  /  x ]. ph )
 )
 
Theoremralsng 3411* Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A  e.  V  ->  ( A. x  e.  { A } ph  <->  ps ) )
 
Theoremrexsng 3412* Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A  e.  V  ->  ( E. x  e.  { A } ph  <->  ps ) )
 
Theoremexsnrex 3413 There is a set being the element of a singleton if and only if there is an element of the singleton. (Contributed by Alexander van der Vekens, 1-Jan-2018.)
 |-  ( E. x  M  =  { x }  <->  E. x  e.  M  M  =  { x } )
 
Theoremralsn 3414* Convert a quantification over a singleton to a substitution. (Contributed by NM, 27-Apr-2009.)
 |-  A  e.  _V   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A. x  e.  { A } ph  <->  ps )
 
Theoremrexsn 3415* Restricted existential quantification over a singleton. (Contributed by Jeff Madsen, 5-Jan-2011.)
 |-  A  e.  _V   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E. x  e.  { A } ph  <->  ps )
 
Theoremeltpg 3416 Members of an unordered triple of classes. (Contributed by FL, 2-Feb-2014.) (Proof shortened by Mario Carneiro, 11-Feb-2015.)
 |-  ( A  e.  V  ->  ( A  e.  { B ,  C ,  D }  <->  ( A  =  B  \/  A  =  C  \/  A  =  D ) ) )
 
Theoremeltpi 3417 A member of an unordered triple of classes is one of them. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  ( A  e.  { B ,  C ,  D }  ->  ( A  =  B  \/  A  =  C  \/  A  =  D ) )
 
Theoremeltp 3418 A member of an unordered triple of classes is one of them. Special case of Exercise 1 of [TakeutiZaring] p. 17. (Contributed by NM, 8-Apr-1994.) (Revised by Mario Carneiro, 11-Feb-2015.)
 |-  A  e.  _V   =>    |-  ( A  e.  { B ,  C ,  D }  <->  ( A  =  B  \/  A  =  C  \/  A  =  D ) )
 
Theoremdftp2 3419* Alternate definition of unordered triple of classes. Special case of Definition 5.3 of [TakeutiZaring] p. 16. (Contributed by NM, 8-Apr-1994.)
 |- 
 { A ,  B ,  C }  =  { x  |  ( x  =  A  \/  x  =  B  \/  x  =  C ) }
 
Theoremnfpr 3420 Bound-variable hypothesis builder for unordered pairs. (Contributed by NM, 14-Nov-1995.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x { A ,  B }
 
Theoremralprg 3421* Convert a quantification over a pair to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ( x  =  B  ->  (
 ph 
 <->  ch ) )   =>    |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A. x  e.  { A ,  B } ph  <->  ( ps  /\  ch ) ) )
 
Theoremrexprg 3422* Convert a quantification over a pair to a disjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ( x  =  B  ->  (
 ph 
 <->  ch ) )   =>    |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( E. x  e.  { A ,  B } ph  <->  ( ps  \/  ch ) ) )
 
Theoremraltpg 3423* Convert a quantification over a triple to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ( x  =  B  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  C  ->  (
 ph 
 <-> 
 th ) )   =>    |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A. x  e.  { A ,  B ,  C } ph 
 <->  ( ps  /\  ch  /\ 
 th ) ) )
 
Theoremrextpg 3424* Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ( x  =  B  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  C  ->  (
 ph 
 <-> 
 th ) )   =>    |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( E. x  e.  { A ,  B ,  C } ph 
 <->  ( ps  \/  ch  \/  th ) ) )
 
Theoremralpr 3425* Convert a quantification over a pair to a conjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ( x  =  B  ->  ( ph  <->  ch ) )   =>    |-  ( A. x  e. 
 { A ,  B } ph  <->  ( ps  /\  ch ) )
 
Theoremrexpr 3426* Convert an existential quantification over a pair to a disjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ( x  =  B  ->  ( ph  <->  ch ) )   =>    |-  ( E. x  e. 
 { A ,  B } ph  <->  ( ps  \/  ch ) )
 
Theoremraltp 3427* Convert a quantification over a triple to a conjunction. (Contributed by NM, 13-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  ( x  =  B  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  C  ->  (
 ph 
 <-> 
 th ) )   =>    |-  ( A. x  e.  { A ,  B ,  C } ph  <->  ( ps  /\  ch 
 /\  th ) )
 
Theoremrextp 3428* Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  ( x  =  B  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  C  ->  (
 ph 
 <-> 
 th ) )   =>    |-  ( E. x  e.  { A ,  B ,  C } ph  <->  ( ps  \/  ch 
 \/  th ) )
 
Theoremsbcsng 3429* Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  A. x  e.  { A } ph ) )
 
Theoremnfsn 3430 Bound-variable hypothesis builder for singletons. (Contributed by NM, 14-Nov-1995.)
 |-  F/_ x A   =>    |-  F/_ x { A }
 
Theoremcsbsng 3431 Distribute proper substitution through the singleton of a class. (Contributed by Alan Sare, 10-Nov-2012.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_
 { B }  =  { [_ A  /  x ]_ B } )
 
Theoremdisjsn 3432 Intersection with the singleton of a non-member is disjoint. (Contributed by NM, 22-May-1998.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof shortened by Wolf Lammen, 30-Sep-2014.)
 |-  ( ( A  i^i  { B } )  =  (/) 
 <->  -.  B  e.  A )
 
Theoremdisjsn2 3433 Intersection of distinct singletons is disjoint. (Contributed by NM, 25-May-1998.)
 |-  ( A  =/=  B  ->  ( { A }  i^i  { B } )  =  (/) )
 
Theoremdisjpr2 3434 The intersection of distinct unordered pairs is disjoint. (Contributed by Alexander van der Vekens, 11-Nov-2017.)
 |-  ( ( ( A  =/=  C  /\  B  =/=  C )  /\  ( A  =/=  D  /\  B  =/=  D ) )  ->  ( { A ,  B }  i^i  { C ,  D } )  =  (/) )
 
Theoremsnprc 3435 The singleton of a proper class (one that doesn't exist) is the empty set. Theorem 7.2 of [Quine] p. 48. (Contributed by NM, 5-Aug-1993.)
 |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
 
Theoremr19.12sn 3436* Special case of r19.12 2422 where its converse holds. (Contributed by NM, 19-May-2008.) (Revised by Mario Carneiro, 23-Apr-2015.)
 |-  A  e.  _V   =>    |-  ( E. x  e.  { A } A. y  e.  B  ph  <->  A. y  e.  B  E. x  e.  { A } ph )
 
Theoremrabsn 3437* Condition where a restricted class abstraction is a singleton. (Contributed by NM, 28-May-2006.)
 |-  ( B  e.  A  ->  { x  e.  A  |  x  =  B }  =  { B } )
 
Theoremrabrsndc 3438* A class abstraction over a decidable proposition restricted to a singleton is either the empty set or the singleton itself. (Contributed by Jim Kingdon, 8-Aug-2018.)
 |-  A  e.  _V   &    |- DECID  ph   =>    |-  ( M  =  { x  e.  { A }  |  ph }  ->  ( M  =  (/)  \/  M  =  { A } )
 )
 
Theoremeuabsn2 3439* Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |-  ( E! x ph  <->  E. y { x  |  ph }  =  { y }
 )
 
Theoremeuabsn 3440 Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by NM, 22-Feb-2004.)
 |-  ( E! x ph  <->  E. x { x  |  ph }  =  { x }
 )
 
Theoremreusn 3441* A way to express restricted existential uniqueness of a wff: its restricted class abstraction is a singleton. (Contributed by NM, 30-May-2006.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)
 |-  ( E! x  e.  A  ph  <->  E. y { x  e.  A  |  ph }  =  { y } )
 
Theoremabsneu 3442 Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.)
 |-  ( ( A  e.  V  /\  { x  |  ph
 }  =  { A } )  ->  E! x ph )
 
Theoremrabsneu 3443 Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.) (Revised by Mario Carneiro, 23-Dec-2016.)
 |-  ( ( A  e.  V  /\  { x  e.  B  |  ph }  =  { A } )  ->  E! x  e.  B  ph )
 
Theoremeusn 3444* Two ways to express " A is a singleton." (Contributed by NM, 30-Oct-2010.)
 |-  ( E! x  x  e.  A  <->  E. x  A  =  { x } )
 
Theoremrabsnt 3445* Truth implied by equality of a restricted class abstraction and a singleton. (Contributed by NM, 29-May-2006.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
 |-  B  e.  _V   &    |-  ( x  =  B  ->  (
 ph 
 <->  ps ) )   =>    |-  ( { x  e.  A  |  ph }  =  { B }  ->  ps )
 
Theoremprcom 3446 Commutative law for unordered pairs. (Contributed by NM, 5-Aug-1993.)
 |- 
 { A ,  B }  =  { B ,  A }
 
Theorempreq1 3447 Equality theorem for unordered pairs. (Contributed by NM, 29-Mar-1998.)
 |-  ( A  =  B  ->  { A ,  C }  =  { B ,  C } )
 
Theorempreq2 3448 Equality theorem for unordered pairs. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  =  B  ->  { C ,  A }  =  { C ,  B } )
 
Theorempreq12 3449 Equality theorem for unordered pairs. (Contributed by NM, 19-Oct-2012.)
 |-  ( ( A  =  C  /\  B  =  D )  ->  { A ,  B }  =  { C ,  D }
 )
 
Theorempreq1i 3450 Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)
 |-  A  =  B   =>    |-  { A ,  C }  =  { B ,  C }
 
Theorempreq2i 3451 Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)
 |-  A  =  B   =>    |-  { C ,  A }  =  { C ,  B }
 
Theorempreq12i 3452 Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)
 |-  A  =  B   &    |-  C  =  D   =>    |- 
 { A ,  C }  =  { B ,  D }
 
Theorempreq1d 3453 Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  { A ,  C }  =  { B ,  C }
 )
 
Theorempreq2d 3454 Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  { C ,  A }  =  { C ,  B }
 )
 
Theorempreq12d 3455 Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  { A ,  C }  =  { B ,  D } )
 
Theoremtpeq1 3456 Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
 |-  ( A  =  B  ->  { A ,  C ,  D }  =  { B ,  C ,  D } )
 
Theoremtpeq2 3457 Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
 |-  ( A  =  B  ->  { C ,  A ,  D }  =  { C ,  B ,  D } )
 
Theoremtpeq3 3458 Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
 |-  ( A  =  B  ->  { C ,  D ,  A }  =  { C ,  D ,  B } )
 
Theoremtpeq1d 3459 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  { A ,  C ,  D }  =  { B ,  C ,  D } )
 
Theoremtpeq2d 3460 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  { C ,  A ,  D }  =  { C ,  B ,  D } )
 
Theoremtpeq3d 3461 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  { C ,  D ,  A }  =  { C ,  D ,  B } )
 
Theoremtpeq123d 3462 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   &    |-  ( ph  ->  E  =  F )   =>    |-  ( ph  ->  { A ,  C ,  E }  =  { B ,  D ,  F } )
 
Theoremtprot 3463 Rotation of the elements of an unordered triple. (Contributed by Alan Sare, 24-Oct-2011.)
 |- 
 { A ,  B ,  C }  =  { B ,  C ,  A }
 
Theoremtpcoma 3464 Swap 1st and 2nd members of an undordered triple. (Contributed by NM, 22-May-2015.)
 |- 
 { A ,  B ,  C }  =  { B ,  A ,  C }
 
Theoremtpcomb 3465 Swap 2nd and 3rd members of an undordered triple. (Contributed by NM, 22-May-2015.)
 |- 
 { A ,  B ,  C }  =  { A ,  C ,  B }
 
Theoremtpass 3466 Split off the first element of an unordered triple. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |- 
 { A ,  B ,  C }  =  ( { A }  u.  { B ,  C }
 )
 
Theoremqdass 3467 Two ways to write an unordered quadruple. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |-  ( { A ,  B }  u.  { C ,  D } )  =  ( { A ,  B ,  C }  u.  { D } )
 
Theoremqdassr 3468 Two ways to write an unordered quadruple. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |-  ( { A ,  B }  u.  { C ,  D } )  =  ( { A }  u.  { B ,  C ,  D } )
 
Theoremtpidm12 3469 Unordered triple  { A ,  A ,  B } is just an overlong way to write  { A ,  B }. (Contributed by David A. Wheeler, 10-May-2015.)
 |- 
 { A ,  A ,  B }  =  { A ,  B }
 
Theoremtpidm13 3470 Unordered triple  { A ,  B ,  A } is just an overlong way to write  { A ,  B }. (Contributed by David A. Wheeler, 10-May-2015.)
 |- 
 { A ,  B ,  A }  =  { A ,  B }
 
Theoremtpidm23 3471 Unordered triple  { A ,  B ,  B } is just an overlong way to write  { A ,  B }. (Contributed by David A. Wheeler, 10-May-2015.)
 |- 
 { A ,  B ,  B }  =  { A ,  B }
 
Theoremtpidm 3472 Unordered triple  { A ,  A ,  A } is just an overlong way to write  { A }. (Contributed by David A. Wheeler, 10-May-2015.)
 |- 
 { A ,  A ,  A }  =  { A }
 
Theoremtppreq3 3473 An unordered triple is an unordered pair if one of its elements is identical with another element. (Contributed by Alexander van der Vekens, 6-Oct-2017.)
 |-  ( B  =  C  ->  { A ,  B ,  C }  =  { A ,  B }
 )
 
Theoremprid1g 3474 An unordered pair contains its first member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.)
 |-  ( A  e.  V  ->  A  e.  { A ,  B } )
 
Theoremprid2g 3475 An unordered pair contains its second member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.)
 |-  ( B  e.  V  ->  B  e.  { A ,  B } )
 
Theoremprid1 3476 An unordered pair contains its first member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 5-Aug-1993.)
 |-  A  e.  _V   =>    |-  A  e.  { A ,  B }
 
Theoremprid2 3477 An unordered pair contains its second member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 5-Aug-1993.)
 |-  B  e.  _V   =>    |-  B  e.  { A ,  B }
 
Theoremprprc1 3478 A proper class vanishes in an unordered pair. (Contributed by NM, 5-Aug-1993.)
 |-  ( -.  A  e.  _V 
 ->  { A ,  B }  =  { B } )
 
Theoremprprc2 3479 A proper class vanishes in an unordered pair. (Contributed by NM, 22-Mar-2006.)
 |-  ( -.  B  e.  _V 
 ->  { A ,  B }  =  { A } )
 
Theoremprprc 3480 An unordered pair containing two proper classes is the empty set. (Contributed by NM, 22-Mar-2006.)
 |-  ( ( -.  A  e.  _V  /\  -.  B  e.  _V )  ->  { A ,  B }  =  (/) )
 
Theoremtpid1 3481 One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  A  e.  _V   =>    |-  A  e.  { A ,  B ,  C }
 
Theoremtpid2 3482 One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  B  e.  _V   =>    |-  B  e.  { A ,  B ,  C }
 
Theoremtpid3g 3483 Closed theorem form of tpid3 3484. (Contributed by Alan Sare, 24-Oct-2011.)
 |-  ( A  e.  B  ->  A  e.  { C ,  D ,  A }
 )
 
Theoremtpid3 3484 One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  C  e.  _V   =>    |-  C  e.  { A ,  B ,  C }
 
Theoremsnnzg 3485 The singleton of a set is not empty. (Contributed by NM, 14-Dec-2008.)
 |-  ( A  e.  V  ->  { A }  =/=  (/) )
 
Theoremsnmg 3486* The singleton of a set is inhabited. (Contributed by Jim Kingdon, 11-Aug-2018.)
 |-  ( A  e.  V  ->  E. x  x  e. 
 { A } )
 
Theoremsnnz 3487 The singleton of a set is not empty. (Contributed by NM, 10-Apr-1994.)
 |-  A  e.  _V   =>    |-  { A }  =/= 
 (/)
 
Theoremsnm 3488* The singleton of a set is inhabited. (Contributed by Jim Kingdon, 11-Aug-2018.)
 |-  A  e.  _V   =>    |-  E. x  x  e.  { A }
 
Theoremprmg 3489* A pair containing a set is inhabited. (Contributed by Jim Kingdon, 21-Sep-2018.)
 |-  ( A  e.  V  ->  E. x  x  e. 
 { A ,  B } )
 
Theoremprnz 3490 A pair containing a set is not empty. (Contributed by NM, 9-Apr-1994.)
 |-  A  e.  _V   =>    |-  { A ,  B }  =/=  (/)
 
Theoremprm 3491* A pair containing a set is inhabited. (Contributed by Jim Kingdon, 21-Sep-2018.)
 |-  A  e.  _V   =>    |-  E. x  x  e.  { A ,  B }
 
Theoremprnzg 3492 A pair containing a set is not empty. (Contributed by FL, 19-Sep-2011.)
 |-  ( A  e.  V  ->  { A ,  B }  =/=  (/) )
 
Theoremtpnz 3493 A triplet containing a set is not empty. (Contributed by NM, 10-Apr-1994.)
 |-  A  e.  _V   =>    |-  { A ,  B ,  C }  =/= 
 (/)
 
Theoremsnss 3494 The singleton of an element of a class is a subset of the class. Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 5-Aug-1993.)
 |-  A  e.  _V   =>    |-  ( A  e.  B 
 <->  { A }  C_  B )
 
Theoremeldifsn 3495 Membership in a set with an element removed. (Contributed by NM, 10-Oct-2007.)
 |-  ( A  e.  ( B  \  { C }
 ) 
 <->  ( A  e.  B  /\  A  =/=  C ) )
 
Theoremeldifsni 3496 Membership in a set with an element removed. (Contributed by NM, 10-Mar-2015.)
 |-  ( A  e.  ( B  \  { C }
 )  ->  A  =/=  C )
 
Theoremneldifsn 3497  A is not in  ( B 
\  { A }
). (Contributed by David Moews, 1-May-2017.)
 |- 
 -.  A  e.  ( B  \  { A }
 )
 
Theoremneldifsnd 3498  A is not in  ( B 
\  { A }
). Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  -.  A  e.  ( B  \  { A } ) )
 
Theoremrexdifsn 3499 Restricted existential quantification over a set with an element removed. (Contributed by NM, 4-Feb-2015.)
 |-  ( E. x  e.  ( A  \  { B } ) ph  <->  E. x  e.  A  ( x  =/=  B  /\  ph ) )
 
Theoremsnssg 3500 The singleton of an element of a class is a subset of the class. Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 22-Jul-2001.)
 |-  ( A  e.  V  ->  ( A  e.  B  <->  { A }  C_  B ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10124
  Copyright terms: Public domain < Previous  Next >